Diophantine approximation of the exponential function and Sondow?s Conjecture

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• 作者：Bruce C. Berndt ; Sun Kim ; Alex ; ru Zaharescu
• 关键词：primary ; 11J70 ; secondary ; 11J17 ; 11J68
• 刊名：Advances in Mathematics
• 出版年：2013
• 出版时间：25 November, 2013
• 年：2013
• 卷：248
• 期：Complete
• 页码：1298-1331
• 全文大小：408 K

文摘

We begin by examining a hitherto unexamined partial manuscript by Ramanujan on the diophantine approximation of published with his lost notebook. This diophantine approximation is then used to study the problem of how often the partial Taylor series sums of e coincide with the convergents of the (simple) continued fraction of e. We then develop a p-adic analysis of the denominators of the convergents of e and prove a conjecture of J. Sondow that there are only two instances when the convergents of the continued fraction of e coalesce with partial sums of e. We conclude with open questions about the zeros of certain p-adic functions naturally occurring in our proofs.